A store sells almonds for [tex]$\$ 7$[/tex] per pound, cashews for [tex]$\$ 10$[/tex] per pound, and walnuts for [tex]$\$ 12$[/tex] per pound. A customer buys 12 pounds of mixed nuts consisting of almonds, cashews, and walnuts for [tex]$\$ 118$[/tex]. The customer buys 2 more pounds of walnuts than cashews. The matrix below represents this situation:

[tex]
\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}
[/tex]

If the reduced row echelon form of this matrix represents the amount of each type of nut the customer buys, which statement is a possible interpretation of the results?

A. The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.

B. The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews.

C. The customer buys 0.5 more pound of walnuts than almonds and 2.5 more pounds of almonds than cashews.

D. The customer buys 6.5 more pounds of walnuts than almonds and 8.5 more pounds of almonds than cashews.

Let's go through the process step-by-step to figure out how many pounds of each type of nut the customer buys.

### Equation Setup:

1. Variables:
- Let [tex]$ x $[/tex] be the number of pounds of cashews.
- Let [tex]$ y $[/tex] be the number of pounds of almonds.
- Let [tex]$ z $[/tex] be the number of pounds of walnuts.

2. Equations based on the problem:

- Equation 1: From "the customer buys 2 more pounds of walnuts than cashews," we get:
[tex]\[
z = x + 2
\][/tex]

- Equation 2: From the total cost of the nuts being $118:
[tex]\[
7y + 10x + 12z = 118
\][/tex]

- Equation 3: From the total weight of the mixed nuts being 12 pounds:
[tex]\[
x + y + z = 12
\][/tex]

### Solving the Equations:

1. Substitute [tex]$ z $[/tex] from Equation 1:

Use [tex]$ z = x + 2 $[/tex] and substitute it in Equations 2 and 3.

2. Substitute in Equation 3:
[tex]\[
x + y + (x + 2) = 12
\][/tex]

Simplify:
[tex]\[
2x + y + 2 = 12
\][/tex]

Subtract 2 from both sides:
[tex]\[
2x + y = 10
\][/tex]

So:
[tex]\[
y = 10 - 2x \tag{Equation 4}
\][/tex]

3. Substitute in Equation 2:
[tex]\[
7y + 10x + 12(x + 2) = 118
\][/tex]

Substitute [tex]$ y = 10 - 2x $[/tex] from Equation 4:
[tex]\[
7(10 - 2x) + 10x + 12(x + 2) = 118
\][/tex]

Simplify:
[tex]\[
70 - 14x + 10x + 12x + 24 = 118
\][/tex]

Combine terms:
[tex]\[
70 + 24 + 8x = 118
\][/tex]

Simplify:
[tex]\[
94 + 8x = 118
\][/tex]

Subtract 94 from both sides:
[tex]\[
8x = 24
\][/tex]

Divide by 8:
[tex]\[
x = 3
\][/tex]

4. Find [tex]$ y $[/tex] and [tex]$ z $[/tex]:

Using [tex]$ x = 3 $[/tex] in Equation 4:
[tex]\[
y = 10 - 2(3)
\][/tex]

[tex]\[
y = 10 - 6
\][/tex]

[tex]\[
y = 4
\][/tex]

Use [tex]$ x = 3 $[/tex] in Equation 1 to find [tex]$ z $[/tex]:
[tex]\[
z = x + 2
\][/tex]

[tex]\[
z = 3 + 2
\][/tex]

[tex]\[
z = 5
\][/tex]

### Conclusion:

- The customer buys 3 pounds of cashews ([tex]$x = 3$[/tex]).
- The customer buys 4 pounds of almonds ([tex]$y = 4$[/tex]).
- The customer buys 5 pounds of walnuts ([tex]$z = 5$[/tex]).

Therefore, a possible interpretation of the situation is that the customer buys 2 more pounds of walnuts than almonds and 1 more pound of almonds than cashews.